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Home Multivariate Data Modeling Neural Networks RBF Neural Networks RBF Network as Kernel Estimator  
See also: RBF network  
RBF Network as Kernel EstimatorRBF neural networks belong to the class of kernel estimation methods. These methods use a weighted sum of a finite set of nonlinear functions Φ(xc_{i}) to approximate an unknown function f(x). The approximation is constructed from the data samples presented to the network using the following equation:
where h is the number of kernel functions, Φ() is the kernel function, x is the input vector, c is a vector which represents the center of the kernel function in the ndimensional space, and w_{i} are the coefficients to adapt the approximating function f(x). If these kernel functions are mapped to a neuralnetwork architecture, a threelayered network can be constructed where eachhidden node is represented by a single kernel function and the coefficients w_{i} represent the weights of the outputlayer. The type of each kernel function can be chosen out of a large class of functions and in fact, it has been shown more recently that an arbitrary nonlinearity is sufficient for representing any functional relationship by a neural network. Gaussian kernel functions are widely used throughout the literature. It has been shown that a small modification to the Gaussian kernel function improves the performance of RBFnetworks for classification tasks:
When R equals 0, the kernel function is the classical Gaussian function (see figure below). A large R creates a flat top of the kernel which more and more approaches the form of a cylinder with increasing R.
The output layer of an RBF network combines the kernel function of all hidden neurons with a linearweighted sum of these functions. Depending on various parameters, the response of the network can assume virtually all thinkable shapes. Several possible response functions obtained from a network with five hidden neurons by varying the S and the R parameters are displayed below.


Home Multivariate Data Modeling Neural Networks RBF Neural Networks RBF Network as Kernel Estimator 